Thermochemistry of Racemic and Enantiopure Organic Crystals for

Jul 18, 2017 - The separation of an enantiomer from a racemic mixture is of primary relevance to the pharmaceutical industry. The thermochemical prope...
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Thermochemistry of Racemic and Enantiopure Organic Crystals for Predicting Enantiomer Separation Hannes K. Buchholz,†,# Rebecca K. Hylton,†,‡,# Jan Gerit Brandenburg,‡ Andreas Seidel-Morgenstern,† Heike Lorenz,† Matthias Stein,*,† and Sarah L. Price*,‡ †

Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstrasse 1, 39106 Magdeburg, Germany Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, U.K.

‡

S Supporting Information *

ABSTRACT: The separation of an enantiomer from a racemic mixture is of primary relevance to the pharmaceutical industry. The thermochemical properties of organic enantiopure and racemate crystals can be exploited to design an enantioselective crystallization process. The thermodynamic difference between the two crystal forms is accessible by two cycles which give the eutectic composition in solution. The “sublimation cycle” requires calculating the lattice energy and phonon frequencies of the crystal structures. Experimental results from heat capacity and other thermodynamic measurements of enantiopure and racemic crystals are compared with a variety of molecular and crystal structure-based calculations. This is done for three prototypes of pharmaceutical-like molecules with different degrees of molecular flexibility. Differences in crystal packing result in varying temperature-dependent heat capacities and affect the sublimation thermodynamics, relative solubility, and eutectic composition. Many simplifying assumptions about the thermodynamics and solubilities of the racemic and enantiopure crystals are critically evaluated. We show that calculations and experimental information using the sublimation cycle can guide the design of processes to resolve enantiomers by crystallization. step crystallization process.7 Polymorphism and the formation of solvates further complicate advanced process design. In early stages of drug development, it is desirable to have computational methods to support, or possibly replace, many experimental investigations. The difference in solution free energies ΔRS−S ΔGsol determines the enantiomeric enrichment. ΔGsol can, in principle, be obtained from two hypothetical thermodynamic cycles8,9 (see Figure 1): (i) transfer of the molecule from crystal to the supercooled melt and then exchange of molecules between the melt and solvent, and (ii) transfer of the molecule from crystal to vapor and then into solution. Cycle (i), the “melt cycle”, requires knowledge of the melting temperature and enthalpy of melting for determining the Gibbs free energy of melting via ΔGmelt = ΔHmelt − T ΔHmelt/Tmelt plus thermal contributions of the solid and the supercooled melt (see Figure 1). This approach is used by the general solubility equation (GSE),10−12 other QSPR methods,13 and by activity coefficient based models.14 Several organic compounds already decompose before or during melting, which poses a severe restriction on its

1. INTRODUCTION New pharmaceuticals are predominantly chiral compounds but often are synthesized as racemates, i.e., an equimolar mixture of two enantiomers.1 Despite identical chemical structures, the two mirror images display different pharmacological, toxicological, and pharmacokinetic properties. Chiral separation of these compounds or proof of harmlessness of the inactive enantiomer is now required for drug product approval2 and is gaining importance also in the agrochemical sector.3 There are different processes to separate enantiomers, but, if possible, crystallization is usually the preferred method for manufacture.4 Crystallization-based separation of enantiomers leads to pure crystalline enantiomers, and recent computational estimates suggest5 that it may be applicable to more systems if the molecular basis were fully understood. Such an enantioselective crystallization process depends on the ternary phase diagram between crystalline enantiomer, racemate, and solvent when crystallization is driven by thermodynamic control. The key parameters to a successful process are the solubilities of the pure enantiomer and the racemate, the composition of the eutectic system in the given solvent (mixture), and temperature. The eutectic composition, defined as the maximum solubility of one enantiomer in the presence of the second (see eu eu Supporting Information, Figure S1), xeu = xeu S /(xS + xR ) can be 6,7 a function of temperature, T, and solvent. In some cases shifts in the composition at the eutectic (“eutectic shifts”) can vary sufficiently with temperature to allow separation by a two© XXXX American Chemical Society

Received: April 24, 2017 Revised: July 3, 2017 Published: July 18, 2017 A

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temperature-dependent isobaric heat capacity difference between the gas (g) and solid (s). The term Hcorr is the sum of the zero-point energies (ZPEs) and the integrals of CP and is often referred to as the “thermal correction”. Thus, calculations of the solubility will not only be dependent on the accuracy of the lattice energy calculations, but also on that of other contributions to the thermodynamic cycles.19 The molecules investigated in this study (Figure 2) were chosen to represent prototypes of frequently occurring organic

Figure 2. Molecular structures of the three representative chiral molecules: (left) 3,6-dimethyl-1,4-dioxane-2,5-dione (lactide C6H8O4), (middle) naproxen C14H14O3, (right) 3-chloromandelic acid (3ClMA, C8H7O3Cl). The most flexible torsion angles are marked.

Figure 1. Two thermodynamic cycles to obtain the free energy of solution ΔGsol. The melt cycle relies on the free energy of melting, ΔGmelt, and on the free energy of mixing ΔGE = RT ln γ. The sublimation cycle uses the free energies of sublimation ΔGsubl, and the solvation free energy ΔGsolv. Solvent-dependent properties are in red.

crystalline compound classes: an almost rigid molecule with no hydrogen bonding capability (lactide), a drug molecule with intermediate flexibility but only one hydrogen bonding group (naproxen), and a molecule with considerable flexibility and competing hydrogen bond donors close to the chiral center (3chloromandelic acid). The lactide is an internal cyclic ester of lactic acid. Both the racemic molecular compound and the (S)-enantiomer are used in the preparation of polylactide,24,25 a biologically decomposable polymer with many complex properties that make it ideal for medical applications.26 Naproxen is marketed in the enantiopure form as a nonsteroidal anti-inflammatory drug, although the racemic form is more stable.27,28 3-Chloromandelic acid (3ClMA) is used as an intermediate for pharmaceutical products and represents the mandelic acid family of prototypical chiral molecules.29 It has complex polymorphic crystallization behavior that defies many current assumptions.29,30 Temperature-dependent heat capacities for the enantiopure and racemic crystals were determined experimentally and by computational methods for these substances. This experimental comparison is rare, with most examples being zwitterionic amino acids.31−37 Assuming that the eutectic composition is under thermodynamic control,6,38,39 we use a variety of experimental thermodynamic measurements and the “sublimation cycle” to evaluate whether current computational approaches can predict the eutectic composition. We critically assess the underlying assumptions that are being made to make the calculations feasible. The dominant contribution of breaking up the crystal lattice is the lattice energy, Elatt. The lattice energy has been widely used for understanding and modeling the organic solid state,40−44 but improvements in modeling Elatt show that relative finite temperature thermodynamic properties, arising from the molecular motion, are relevant. We are assessing first the separated model ψmol for which molecular vibrations are assumed to be the same in the gas and both crystals, and so the rigid-body lattice modes can be used.45,46 This requires electronic structure calculations on the isolated molecule only, as these can be used to define an intermolecular potential for modeling the crystals. The second recognizes that crystal packing can modify both molecular and phonon lattice modes.47 Calculating the coupled vibrational modes is dependent on the potential energy surface for nuclear

applicability as the melting properties can only be predicted computationally by group contribution or QSPR methods which cannot distinguish between polymorphs or chiral molecules.15,16 The other major term, the partial molar Gibbs free energy of mixing the solvent with the supercooled melt, is accessible by correlative activity coefficient and predictive based models (gE-models)14,17,18 Cycle (ii), the “sublimation cycle”, the focus of this paper, approximates the solubility differences by taking the molecule from the crystal to the gas phase and then into solution. Both cycles consider properties from the crystals (melting or sublimation free energy) and are thus dependent on the crystal structures of the racemate and enantiomer. The sublimation cycle is preferred as it makes use of the computationally accessible thermodynamics states from the gas and the crystalline solid. The “sublimation cycle” has been evaluated for the computational prediction of the solubility19 and sublimation free energies of organic crystals.20 Here, we evaluate the sublimation cycle to obtain differences in solubilities of the racemic and enantiopure crystals and to assess the relevance of differences in thermodynamic terms which are often taken to be negligible. In the sublimation cycle, the mole fraction solubility, x, is directly related to the Gibbs free energy of solution21−23 and can be calculated from the sublimation and solvation free energies (lower part of Figure 1). −RT ln x = ΔGsol = ΔGsubl + ΔGsolv

(1)

Assuming that there is no polymorphic phase transition between 0 K and the temperature of interest, T, (e.g., the solution temperature), the sublimation free energy is given by ΔGsubl = ΔHsubl − ΔTSsubl = −E latt + Hcorr − T ΔSsubl g s with Hcorr = (EZPE − EZPE )+ T

and T ΔSsubl = T

∫T′=0

T

∫T′=0 ΔCPg−s(T′) dT′

ΔCPg − s(T ′) dT ′ T′ (2)

where Elatt is the lattice energy, EZPE is the zero-point g s vibrational energy and Cg−s P (T) = CP(T) − CP(T) is the B

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assuming an ideal gas and the Dulong-Petit model of a perfect atomic crystal. 52 This relationship enables sublimation enthalpies to be estimated without the need to calculate the vibrational (phonon) modes of the crystal. However, recently this approximation has been shown to be poor, with errors up to 5 kJ mol−1 (Supporting Information, Figure S3) for small molecular crystals.53−55 However, for comparing the thermodynamic dif ferences between enantiopure and racemic crystals, parts of the thermodynamic quantities of Hcorr are likely to cancel out, with the gas phase contributions to Hcorr (eq 2) being assumed to be identical for both chiral species. 2.2. Lattice and Zero-Point Energies, Heat Capacities, and Entropic Contributions. The lattice energy E latt characterizes a static perfect infinite crystal and an ideal gas phase in its lowest energy conformation, both at 0 K. Taking this as the zero of energy, we can calculate the lattice energy from periodic electronic structure models, ψcrys. Less computationally expensive ψmol calculations further separate the lattice energy into crystal and isolated molecule contributions, providing a separated model:56

motion within the crystal, and hence the use of periodic electronic structure calculations ψcrys. Dispersion corrected density functional methods, which have been widely used for lattice energies, proved to be too computationally demanding to obtain converged phonon frequencies across the Brillouin zone,48,49 so more approximate electronic structure methods have been used. We critically assess the accuracy of the underlying assumptions against carefully determined experimental thermodynamic for chiral compounds, rarely available in early stage drug development. This study indicates that the combination of computation and experimental data can assist the design of enantiomeric separation crystallization processes but also points to the challenges in computational process design.

2. BASIC THERMODYNAMIC MODELS In the following we briefly introduce the framework for the computation of the solid-state thermodynamic quantities and its application within a thermodynamic model to predict the eutectic composition of chiral compound-forming systems in solution. 2.1. The Solution Thermodynamics of Chiral Systems. The eutectic composition xeu can be estimated as a function of the solubility ratio between the racemic compound and the enantiomer α = xRS/xS (see Supporting Information, pp S2− S4)6 via the difference in Gibbs free energy of solution between the racemic compound and the enantiomer, ΔRS‑SΔGsol = ΔGsol,RS − ΔGsol,S, to give 1 1 xeu = = Δ ΔG 1 + α 2/4 1 + 1/4 exp −2 RS−S sol

(

RT

)

E latt = Uinter + ΔE intra

where ΔEintra is the difference in energy for a single molecule in the crystal conformation and in its lowest energy conformation (ΔEintra = 0 for rigid molecules) and is typically less than a few kJ·mol−1 unless intramolecular hydrogen bonding is involved.44 Both ψcrys and ψmol models can be used to calculate the lattice vibrations (phonons) from the second derivatives of the lattice energy, i.e., within the harmonic approximation. In the ψmol model, it is assumed that the molecular modes are unaffected by crystal packing and thus can be estimated from the vibrational frequencies of the isolated molecule and combined with the rigid-body lattice modes. Our ψmol model is further simplified by only using the k = 0 modes of the unit cell, ignoring the difference in size and shape between the racemic and enantiopure crystals. ψcrys generates all the vibrational modes, including the high frequency molecular modes, and calculates the heat capacity, zero-point energies and entropy by integration over the Brillouin zone.45,57,58 Full details of both computational models are given in Supporting Information, pp S4−S5. Both methods give EZPE and heat capacities at constant volume CV. The isobaric and isochoric solid state heat capacities CsP and CsV are related by the bulk modulus, K, and the thermal volume expansion coefficient αT.

(3)

This derivation differs in the reference state used from a previously proposed model (see Supporting Information, pp S4−S5).39,50 The two thermodynamic cycles in Figure 1 can be used to relate ΔRS‑SΔGsol, and hence the eutectic composition (eq 3), to either the difference in free energies of melting, ΔRS‑SΔGmelt, or the difference in free energy of sublimation, ΔRS‑SΔGsubl. Focusing on the sublimation cycle we get a relation between α and the sublimation thermodynamics (Figure 1, eq 1) by ⎛ Δ ΔG ⎞ xRS ≈ exp⎜ −2 RS − S subl ⎟ ⎠ ⎝ xS RT ⎛ Δ ⎛ Δ ΔH ⎞ ΔE ⎞ ≈ exp⎜ −2 RS − S subl ⎟ ≈ exp⎜2 RS − S latt ⎟ ⎝ ⎠ ⎝ ⎠ RT RT

α=

CPs (T ) = C Vs(T ) + TVKαT2 (4)

(7)

If the thermal expansion of the unit cell is neglected, which is consistent with making the harmonic approximation, we can directly compare experimental CsP(T) with computed CsV(T). This approximation is made throughout the paper, but it is tested for naproxen using quasi-harmonic calculations (see Supporting Information, p S29−S32).

In eq 4 several levels of approximations are implicit which minimize the computational effort to calculate relative solubilities: (i) the solutions of enantiomer and racemate are ideal, or so similar that ΔRS‑SΔGsolv = 0, and the solvent is achiral; (ii) entropy contributions are the same for the enantiomer and the racemic compound in the solid phase and the gas; (iii) the thermal correction, Hcorr, (eq 2) is equal for the enantiomer and the racemic compound or even molecule independent, as in the frequently used “2RT approximation”: ΔHsubl(T ) = −E latt − 2RT

(6)

3. MATERIALS AND METHODS 3.1. Enantiopure and Racemic Compounds and Crystal Forms. Samples of (S)-lactide, (RS)-lactide, (S)-naproxen, (RS)naproxen, (R)-3ClMA, (R,S)-3ClMA, and (R)- and (RS)-mandelic acid were available commercially and purified and recrystallized where needed, and confirmed to correspond to single polymorphs (see Supporting Information, p S10 for more information). 3.2. Experimental Methods. Heat Capacity Measurements. Low-temperature heat capacities for enantiopure and racemic naproxen and 3ClMA crystals were measured by means of direct heat pulse calorimetry (DHPC) from 2 to 200 K. A commercially

(5)

This approximation is frequently invoked in the computational prediction of solubilities51 and sublimation enthalpies of organic molecules.20 It arises from the equipartition theorem of internal degrees of freedom for the gas and the crystal, C

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available relaxation calorimeter has been used at the Institute of theoretical Physics of the Goethe University in Frankfurt, the physical property measurement system (PPMS) from Quantum Design. Heat capacities above room temperature were measured using a DSC 111 calorimeter from Setaram.59 Sapphire and benzoic acid were used as reference standards. Average deviations to published data were 1.4% for sapphire60 and 2.6% for benzoic acid.61 Additional differential scanning calorimetry (DSC) measurements of the closely related (R)and (RS)-mandelic acid molecules were performed between 298 and 370 K using the same measurement method as for the other molecules (more details can be found in the Supporting Information, p S26). Solid State Raman Spectra. Solid-state Raman analysis of the powdered crystals was performed with a commercial MultiRAM spectrometer from Bruker, Germany. The system employs a laser beam at 1064 nm and was operated at 500 mW. The analyses were carried at ambient temperature averaging over 32 scans with a spectral resolution of 1 cm−1 between 10 and 3500 cm−1. Eutectic Composition and Thermochemical Properties. Eutectic compositions in various solvent systems were taken from literature for lactide62,63 and 3ClMA.30,63 and measured here for (S)- and (RS)naproxen (see Supporting Information, Table 28). Additional solubilities were measured in a variety of solvents for lactide to supplement previous measurements. Melting properties, vapor pressures, and corresponding enthalpies and free energies of sublimation were taken from the literature27,30,64,65 where available and are summarized in the Supporting Information, Tables S3−S4. 3.3. Computational Methods. Within the electronic (ψcrys) model we compare thermochemical properties from two approximate dispersion corrected periodic electronic structure methods based on a minimal basis set Hartree−Fock (HF-3c)66,67 and on a tight-binding Hamiltonian (DFTB3-D3).53 HF-3c and DFTB3 calculations were performed with a developer version of CRYSTAL14,68 and DFTB+,69 respectively. In the separated ψmol model, the isolated molecular structures were geometry optimized and molecular vibrations in the rigid rotor harmonic oscillator (RRHO) approximation were calculated at the PBE/def2-TZVP level of theory using the D3 dispersion correction70 with TURBOMOLE V6.4.71 The rigidmolecule lattice frequencies were calculated using DMACRYS.56 The lattice energy was determined using CrystalOptimizer72 allowing the flexible torsion angles in Figure 2 to adapt to the packing forces. In order to compare experimental and computational enthalpic and entropic contributions, a combined “exp/theory” method is constructed, in which experimental heat capacities are interpolated and numerically integrated using spline functions within MATLAB (Mathworks). These can be combined with calculated ideal gas heat capacities from the isolated molecular frequencies and calculated zeropoint energies from ψmol to give Hcorr and sublimation entropies according to eq 2. The free energy of solvation, ΔGsolv, was computed with the thermodynamic model COSMO-RS73 including the distinct flexibility of the three molecules by a Boltzmann weighting of possible conformers. For more details see Supporting Information, pp S35− S36.

number of atoms per molecule. The results are in agreement with previous measurements for the lactide64,74 and naproxen75 and within the 1−5% accuracy of DSC76 heat capacity measurements. The calculated temperature-dependent heat capacities reproduce the temperature dependence up to 200 K and also give the differences in magnitude between the different organic molecules. At higher temperature, the calculations appear to systematically underestimate heat capacities. which was also found for many small molecular crystals.48 A systematic experimental overestimate of high temperature DSC could be ruled out by comparing to benzoic acid and sapphire (see Supporting Information, p S12 and S28 for more information). 4.2. Heat Capacity Differences between Enantiomeric and Racemic Crystals. Since we are focusing on the free energy differences and relative thermodynamic quantities, heat capacity differences between the racemic and enantiopure substances, ΔRS−SCP(T) = CP,RS − CP,S, are discussed. Below 100 K, the experimental heat capacity differences are between −1 and 3 J mol−1 K−1 and distinguishable between the three molecules (Figure 3), changing in sign between molecules and

Figure 3. Low temperature heat capacity differences, ΔRS‑SCP, between racemic and enantiopure crystals. Experimental results (dots) are shown with experimental uncertainty and compared with calculated ΔRS‑SCV values from electronic ψcrys HF-3c (solid lines) and separated ψmol (dashed lines) phonon modes.

with temperature. The naproxen heat capacity difference for example changes sign twice, rising between 150 and 200 K. The heat capacities for naproxen above 150 K have significant experimental error bars due to a loss in thermal coupling as described by the supplier.77 The calculated heat capacities from ψcrys only qualitatively reproduce differences between enantiomer and racemate at the low temperatures, but do show a marked system dependence. The computed harmonic rigidmolecule, mode separated ψmol calculations in Figure 3 show an asymptotic approach to heat capacity differences of zero for all compounds. The heat capacity differences between the racemic and enantiopure substances, ΔRS‑SCP(T), at 298 K are summarized in Table S25 in the Supporting Information. The high temperature heat capacity differences in the range of 300− 370 K in Figure 4 are particularly pronounced for 3ClMA, where the heat capacity of the enantiomer at 298 K is 211.1 ± 1.6 J mol−1 K−1, which is 9.2 ± 2.2 J mol−1 K−1 (4.5%) larger than that of the racemic crystal (201.9 ± 1.5 J mol−1 K−1; see Figure 4). Such a significant heat capacity difference had

4. RESULTS 4.1. Temperature-Dependent Solid State Heat Capacities. Temperature-dependent heat capacity measurements for enantiopure and racemic crystalline powders of the three substances were recorded in temperature ranges from 2 to 202 K and 298 to 410 K (or until melting) and are given in the Supporting Information, Figures S4−S5, and tabulated in comparison with the calculated values (Supporting Information, Tables S13−S22). At very low temperatures the heat capacities follow Debye’s cubic law (CP ∝ T3), then increase linearly with temperature and do not level even approaching the highest temperatures of 400 K. Thus, the assumption of mode saturation in the 2RT correction (eq 3) is not valid for any of the organic molecular crystals. Naproxen has a significantly larger heat capacity than 3ClMA and lactide due to the larger D

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Figure 4. Experimental high temperature heat capacity differences, ΔRS‑SCP, between the racemate and the enantiomer. Computed electronic ψcrys HF-3c ΔRS‑SCV are given as dashed lines in the same color; separated ψmol ΔRS‑SCV are zero (black straight line).

Figure 5. Comparison of the rigid-molecule k = 0 lattice frequencies of the enantiopure and racemic crystals of lactide, 3ClMA, and naproxen with their isolated molecule low frequency modes, to assess the validity of the separation model ψmol. Numerical values are in the Supporting Information (Tables S5−S10).

previously been reported for mandelic acid31 at temperatures that are relevant for a crystallization process (5−70 °C), so we repeated the experiment for (RS)- and (S)-mandelic acid (see Supporting Information, S23−S24 for primary data) to confirm the previous results,31 and found that the heat capacity difference between the enantiopure and racemic crystals increases with temperature for mandelic acid but is almost constant for 3ClMA (Figure 4). At 298 K the lactide shows a small heat capacity difference, with that of the (RS)-lactide being only 0.6 ± 0.5 J·mol−1 K−1 larger than that of the enantiomer, though this value is smaller than a previously published value of 2.8 ± 0.3 J·mol−1 K−1.64,74 The heat capacity differences of naproxen at room temperature are 1.3 ± 4.9 J· mol−1 K−1 and within experimental uncertainty. The periodic electronic structure calculations do show a small difference in the heat capacities at process-relevant temperatures for naproxen but this is independent of temperature. The separated model ψmol gives no heat capacity differences at elevated temperatures for any molecule. This requires a thorough investigation of the assumptions behind current theoretical models. 4.3. Analysis of Molecular and Lattice Vibrations from Theoretical Models and Underlying Assumptions. Integration of lattice vibrational modes (Supporting Information, Tables S8−S10) yields the heat capacities, characteristic for both molecule type and crystal packing. The separated Ψmol model assumes the lattice modes to be sufficiently separated in energy and uncoupled from molecular vibrational modes (see Supporting Information, Tables S5−S7). Hence, the separated model ψmol inevitably predicts that ΔRS‑SCP tends to zero with increasing temperature at about 150 K, well below ambient temperatures (Figure 3) as the molecular modes are assumed to be unaffected by the crystal structure. This does not hold for any of the compounds (see Figure 5). Even for the “rigid” lactide, there are some ring bending modes that are of similar frequencies to the lattice modes. For naproxen, the low frequency molecular modes are rotations of the propionic acid side chain and bending of the naphthyl ring, which is nonplanar in the enantiopure crystal structure.78 The low frequency molecular modes of 3ClMA are rotations of the flexible alphahydroxy acid side chain and out-of-plane vibrations of the chlorine atom. These very low frequency molecular modes may couple with certain lattice modes and affect the heat capacities

at low temperatures (see equations in Supporting Information, S9−S11). All three molecules pack so differently in the racemic and enantiopure structures that the lowest frequency modes are very different in nature. Figure 6 shows the packing differences and the lowest energy lattice modes within the unit cells. (RS)-naproxen has a layered structure, with only weak van-der-Waals forces between the

Figure 6. Packing of the crystals showing the lowest energy lattice modes by vectors in atomic colors representing the amplitude and direction of the vibrations from HF-3c k = 0 electronic structure calculations. The red shading shows the hydrogen-bonded naproxen layer. E

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zigzag layers of hydrogen bonded dimers. The lowest frequency lattice mode corresponds to a relative sliding of the layers with respect to each other. In (S)-naproxen, there are hydrogenbonded chains which rotate relative to each other. In (RS)3ClMA, pairs of hydrogen-bonding layers slide relative to each other, whereas in (S)-3ClMA the lowest mode is predominately that of flexible torsion around Φ1 (Figure 6). The actual frequencies of these modes will be very sensitive to the choice of computational method and its potential energy surface; indeed, the less computationally demanding periodic electronic structure DFTB3-D3 yields significant differences in low temperature heat capacities compared with the HF-3c calculations (Supporting Information, Tables S11−S22 and Figures S4−S5). Comparable errors result from the assumption that the molecule is rigid and the use of an atom−atom intermolecular potential energy surface in the separated ψmol model for flexible molecules such as naproxen and 3ClMA. The results of ψmol and ψcrys are more consistent for the more rigid lactide, which has more isotropic and very weak intermolecular interactions and shows smaller heat capacity differences. Hence, ψmol and ψcrys both fail in quantitatively calculating the low-temperature heat capacity differences and only give qualitative agreement with experiment (Figure 3) for the molecules that are more typical of pharmaceuticals. However, the error introduced by neglecting the coupling of molecular and lattice vibrational modes and using different computational models is very dependent on the molecule and crystal structure. For the high temperature heat capacities, only the high frequency molecular modes are relevant, as the low frequency lattice modes have become saturated. The electronic ψcrys calculations give differences in the high frequency modes that lead to a small difference in heat capacities at process-relevant temperatures for naproxen and 3ClMA but not for the lactide (Figure 4). The electronic calculations, however, systematically underestimate the high temperature heat capacities (Supporting Information, Figure S11−S22) which contributes to the poor predictions of the heat capacity differences (Figure 4). Differences in molecular and lattice vibrations can be obtained from the solid-state Raman spectra79 for the enantiopure and racemic crystals (see Figure 7). For naproxen and 3ClMA, there are detectable differences between the crystal forms in particular at higher frequencies, i.e., the CO around 1600− 1800 cm−1 and O−H around 3000 cm−1. For lactide, enantiopure and racemic crystals display an almost identical Raman spectrum (Figure 7). Differences in the higher frequency molecular modes between the racemic and enantiopure crystals are necessary and a strong pointer for a difference in the heat capacities at process-relevant temperatures. The harmonic approximation is made in all the calculations and is likely to be poor at relevant temperatures for most organic crystals, as this is relatively close to their melting points (Supporting Information, Table S3). The harmonic approximation completely neglects thermal expansion of crystals, which can differ markedly in its anisotropy with the differences in molecular packing. The anisotropic expansion of naproxen has been estimated by a quasi-harmonic approximation (Supporting Information, pp 29−32) and leads to an increase in the room temperature CP (by 7 and 8 J mol−1 K−1 for RS and S respectively) relative to the harmonic underestimate. Using a fully anharmonic model from Molecular Dynamics trajectories

Figure 7. Solid-state Raman spectra of the racemic and enantiopure crystals between 0 and 1800 cm−1 and 2700−3150 cm−1. The inset illustrates the different hydrogen bonding motifs, responsible for the changes in the spectra at higher frequencies, i.e., the CO around 1600−1800 cm−1 and O−H around 3000 cm−1.

for heat capacities would be expected to further improve the agreement with experiment. 4.4. Comparing Thermochemical Properties for Enantiopure and Racemic Crystals from Experiment and Theory. Figure 8 shows the sublimation enthalpies as estimated using the experimental and calculated thermochemical properties (values in Supporting Information, Table S26). The lattice energy is the major contribution to the absolute values of the enthalpies of sublimation and the differences between the racemate and enantiomer when compared to the experimental values, where available (Figure 8). However, the heat capacity and zero-point energy terms are still significant and cannot be neglected in any computational estimates of the sublimation enthalpy. There is a clear correlation between the differences in the thermal corrections with the flexibility and packing in racemic and enantiopure crystal structures. The molecule and crystalspecific phonon frequencies affect the thermal corrections Hcorr (eq 2), and the “2RT-approximation” is as inadequate for these organic crystals as it is for those comprised of smaller molecules (see Supporting Information, Figure S3). For the lactide species, thermal corrections from separated and periodic calculations are very close and the (RS-S) differences are in excellent agreement with experimental heat capacities corrections, although there are errors in the absolute F

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Figure 8. (a) Absolute sublimation enthalpies, ΔHsubl, and (b) sublimation enthalpy differences between the racemate and enantiomer, ΔRS‑SΔHsubl, compared to experiment as colored background: (blue = lactide;65 green = naproxen;27 3ClMA decomposes), with the plain bars corresponding to the separated model and the striped to the electronic model.

Figure 9. Eutectic compositions at 298 K. Calculated lattice energy, enthalpy corrections, and entropy differences (white, light gray, and gray) from ψmol and ψcrys in the sublimation cycle compared with experiments from both cycles (green) and measured solubilities (yellow) in different solvents. The error bar is its variation. Maximum (solid line) and minimum (dashed line) values of xeu in various solvents at 298 K are also given.

CsP (see Supporting Information, Figure S4). However, for naproxen and 3ClMA the different hydrogen bonding in the two crystals and increasing vibrational flexibility lead to larger differences in thermal corrections. The effect of including the coupling between the molecular and lattice modes is relatively small. The magnitude, however, is still comparable to the neglect of thermal expansion of the unit cell, which is estimated for naproxen (Supporting Information, Figure S9) to lead to an error of about 1 kJ mol−1 in Hcorr. The differences between the calculated and experimental heat capacities are a significant source of error in ΔHsubl, up to 2.2 kJ mol−1 for (S)-3ClMA (Supporting Information, Table S27). 4.5. Estimating the Eutectic Composition and Its Temperature and Solvent Dependence. The eutectic composition is the key parameter for chiral separation. It can depend on temperature, the nature of the solvent, or both. Figure 9 compares calculations of xeu from the thermodynamic cycle using ΔRS‑SΔGsubl, assuming that xeu is independent of solvent, with experimental data at 298 K. A comparison is also made with alternative methods of estimating the eutectic composition. Experimentally, xeu can, in principle, be estimated using the “sublimation cycle” or the “melt cycle” (Figure 1). In the latter, the solution free energy is obtained via the supercooled melt (ΔG melt) by using experimental differences in free energies of melting ΔRS-SΔGmelt (Supporting Information, Tables S3 and S29). The melt cycle assumes that the free energy of mixing with the solvent (ΔGE) is the same for both racemic and enantiopure crystals. Thus, the melt cycle is making the same assumption as the sublimation cycle, that the solution nonidealities of both chiral species are the same (ΔRS‑SΔGE = 0). Results for xeu from the melt cycle are comparable to the sublimation cycle for lactide only. For naproxen, where both the racemic and enantiopure crystals have almost the same melting temperature and enthalpies, the

melt cycle significantly underestimates the Gibbs energy difference and hence the eutectic composition. An alternative approach to estimating xeu, concentrates on the solvent, and uses experimentally measured solubilities of enantiomer and racemate in a variety of solvents (Supporting Information, Table S28) to obtain ΔGsol. If the molecular solvation free energies are calculated using COSMO-RS73 at infinite dilution, and assuming ideal solutions, then ΔGsubl,solub, can be recursively calculated via the sublimation cycle from solubilities (see Supporting Information, Table S29). A cross check on this procedure is that the estimates of ΔRS‑SΔGsubl,solub are in good agreement with the free energies from available vapor pressure measurements (Supporting Information, Table S29) differing only by 0.1 kJ mol−1 for lactide and 1.4 kJ mol−1 for naproxen. Experimental solubilities to estimate xeu (Figure 9 yellow bar) give a variation with solvent which is of the same order of magnitude as the variation of xeu in different solvents at 298 K for lactide, while it is overestimated for naproxen and underestimated for 3ClMA. This shows that there is a significant solvent effect on the eutectic composition at a given temperature because the solutions of the three molecules involved are not ideal, and that this nonideality differs between the racemic and enantiopure solutions. The accuracy of computational predictions of xeu via the sublimation cycle is critically dependent on the accuracy of lattice energy differences. They systematically overestimate the eutectic compositions, as previously found for amino acids in water.39 The enthalpy correction, ΔHcorr, only contributes significantly to xeu in the case of 3ClMA, but entropic G

DOI: 10.1021/acs.cgd.7b00582 Cryst. Growth Des. XXXX, XXX, XXX−XXX

Crystal Growth & Design

Article

differences account for up to 4 kJ·mol−1 to ΔRS‑SΔGsubl and thus affect xeu for all molecules. Direct use of the sublimation cycle when sublimation data are available gives very similar estimates of xeu (Figure 9). The combination of electronic structure calculations and experimental solubilities “exp/theory” (black bar in Figure 9) to yield ΔRS‑SΔGsubl,exp/theory, gives the best estimate of xeu and correlates well with experimental sublimation free energies27,65 when using the ψcrys model. The observed temperature dependence of xeu (Supporting Information, Table S28) cannot be accounted for by the calculations, but can be estimated by numerically integrating the measured heat capacity differences between 298 and 318 K. There is a small temperature effect on the Gibbs free energy of sublimation differences ΔT1‑T2ΔRS‑SΔGsubl which slightly shifts the eutectic composition: for lactide and naproxen, xeu decreases by −0.4 and −0.9 mol % and increases by 0.6 mol % for 3ClMA. In the experiment, these values are dependent on solvent and vary between 0 and −3.0 for lactide and 0 and 1.3 for 3ClMA. For naproxen, the solvent independent shift is slightly underestimated (−1.2 mol %). Thus, the nature of the solvent not only affects the eutectic composition at a given temperature but may also promote or prevent shifts of xeu with temperature. There are examples in the literature, where xeu is temperature dependent in one solvent and independent in another.7 Unlike 3ClMA, the eutectic composition of mandelic acid is constant at xeu = 69% and shows no variation in different solvents and over a large temperature range.4 This is surprising as mandelic acid displays a difference in heat capacity between the racemic and enantiopure crystal which is dependent on temperature.31 This has to be explained by a cancellation of different effects. We have to conclude that the eutectic composition and its temperature dependence Δxeu(T) is not a mere solid state property. Accurate predictions will have to consider the differences in solvation free energies for the racemic and enantiopure solutions. This implies consideration of the nonideality of the solution in terms of the specific molecule, its solvent, as well as solute−solute interactions.

lead to thermal expansion at finite temperature and solid state heat capacities which may differ between the racemic and enantiopure crystals. The often made assumption that lattice energies and sublimation enthalpies differ by 2RT, so that the molecular motion in crystals can be neglected, is oversimplifying the thermodynamics of industrially important organic crystals. Our results also confirm that assuming the solvation energy is the same for the enantiopure and racemic compound is not always realistic enough for the prediction of eutectic shifts.81 5.2. Computational Challenges. The separation of enantiomers poses a challenge to modern computational methods. When lattice energy calculations show that the racemic crystal is much more stable than the enantiopure, the enantiomers in the racemate cannot be separated, but a simple thermodynamically controlled one-step crystallization will remove the racemate leaving an enantiopure solution.21 In the case of small lattice energy differences, separation of enantiomers by a two-step crystallization process may be exploitable. It is very rare for the molecular chirality to provide a strong preference for a chiral packing in all three directions, such that an enantiopure crystal is significantly more stable than any racemic structure.82 Current state-of-the-art electronic structure methods typically have lattice energy errors in the range of 3−7 kJ·mol−1 for crystals of small organic molecules.83,84 When lattice energy differences are small enough for chiral separation to be feasible,